WordPress failure to update 4.4.x

I have had a problem updating WordPress since the 4.4.0 release. Every time I have tried to update my site, (all versions of WordPress since 4.4.0), the site goes unresponsive and I receive a “This page cannot be found” error. Overwriting the updated site directory with the 4.3.x version that I currently have restores the site, so I am stuck using 4.3.x.

• I have tried manually downloading each update and copying the files over with no luck.
• The built-in updater fails halfway through the update because the site essentially disappears during the update. The Load Balancer in front of my servers does not even see that the web servers are there anymore, after attempting an update.
• I have tried all manner of disabling/deleting plugins, none of which has helped
• I have tried overwriting the WP ca-bundle.crt file with the one from Curl CA Cert
• I even had my host update my servers to the latest CentOS release and OpenSSL release that they use
• In short, I have tried everything short of manually updating each file individually to see which file causes the site to break

After reading the forums for quite some time and attempting almost every suggestion that seemed sound, I am at a loss as to why WordPress updates now cause my site to go offline. Any thoughts from the WordPress Core developers would be greatly appreciated.